4.3 Period Changes and Graphs other Trig Functions Obj: Graph sine and cosine with period changes...

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Transcript of 4.3 Period Changes and Graphs other Trig Functions Obj: Graph sine and cosine with period changes...

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4.3 Period Changes and Graphs other Trig FunctionsObj: Graph sine and cosine with period changesObj: Graph other Trig Functions2 EX: Graph y = 3 2 cos xRef, Amp Yes, -2Per 2 Per 03 2 /2 2 2St.Pt. 0Vert. Shift 3

2 EX: Graph y = 3 2 cos x03 22 2 1 0 -1 0 1-2(1 0 -1 0 1) -2 0 2 0 -2

2 EX: Graph y = 3 2 cos x03 2-2(1 0 -1 0 1) 2 2 -2 0 2 0 -2 +3 +3 +3 +3 +31 3 5 3 1

2 EX: Graph y = 3 2 cos x03 2-2(1 0 -1 0 1) 2 2 -2 0 2 0 -2 +3 +3 +3 +3 +31 3 5 3 1

3 EX: Graph y = sinx1

03 2 -1 2 20 1 0 -1 03 EX: Graph y = sinx1

03 2 -1 2 20 1 0 -1 0 3 EX: Graph y = sinx1

03 2 -1 2 2 0 1 0 -1 0-2/3(0 1 0 -1 0)0 -2/3 0 2/3 0 3 EX: Graph y = sinx1

03 2 -1 2 20 -2/3 0 2/3 0+ + + + + -1/6 5/6 3 EX: Graph y = sinx1

03 2 -1 2 24 EX: Graph y = 4 cos (x )1

07 10 13 16 19 -1 6 6 6 6 61 0 -1 0 1 4 EX: Graph y = 4 cos (x )1

0 7 10 13 16 19 -1 6 6 6 6 61 0 -1 0 1 4 EX: Graph y = 4 cos (x )1

0 7 10 13 16 19 -1 6 6 6 6 6 1 0 -1 0 14(1 0 -1 0 1)4 0 -4 0 4 4 EX: Graph y = 4 cos (x )1

0 7 10 13 16 19 -1 6 6 6 6 65 EX: Graph y = 1 +sin(x + /6) - 2 5 8 11 66 6 6 60 1 0 -1 0(0 1 0 -1 0) 0 0 - 05 EX: Graph y = 1 +sin(x + /6) - 2 5 8 11 66 6 6 6 0 0 - 0+1 +1 +1 +1 +11 1 1 15 EX: Graph y = 1 +sin(x + /6) - 2 5 8 11 66 6 6 6 0 0 - 0+1 +1 +1 +1 +11 1 1 14.1 Period changes in graphs of Sine and CosineOBJ: Find the period for a sine and cosine graphy = d + a(trig b (x + c)a (amplitude) multiply a times (0 |1 0 -1 0 1) Sin|Cos-a Reflectionb (period) 2 b can be factored out ORc (starting point) Set (bx + __) = 0 instead of completing factoring with b d (vertical shift)

DEF: Period of Sine and CosineThe graph of y = sin b x will look like thatof sin x, but with a period of 2 . b Also the graph of y = cos b x looks likethat of y = cos x, but with a period of 2 b 8 EX: Graph y = sin 2xRef.noAmp.1Per. 2/2 = Per. /4St. Pt. 0Vert. Sh.none 8 EX: Graph y = sin 2x 2 3 4 -1 4 4 4 4 0 1 0 -1 0 8 EX: Graph y = sin 2x 2 3 4 -1 4 4 4 4 8 EX: Graph y = sin 2xRef.noAmp.1Per. 2/2 = Per. /4St. Pt. 0Vert. Sh.none 0 1 0 -1 0

1 0 -1 /4 3/4 4/4 EX: Graph y = -2cos 3x EX 9 Graph y = 3 2cos 3x 2 3 4 -1 6 6 6 6 EX: Graph y = -2cos 3x EX 9 Graph y = 3 2cos 3x 2 3 4 -1 6 6 6 61 0 -1 0 1 EX: Graph y = -2cos 3x EX 9 Graph y = 3 2cos 3x 2 3 4 -1 6 6 6 6-2(1 0 -1 0 1)-2 0 2 0 -2 2 3 4 -1 6 6 6 6-2 0 2 0 -2+3 +3 +3 +3 +31 3 5 3 1 10 EX: Graph y = 2cos3(x+) 3 10 EX: Graph y = 2cos3(x+) 3 -2 - 2 6 6 6 6 10 EX: Graph y = 2cos3(x+) 3 -2 - 2 6 6 6 6 10 EX: Graph y = 2cos3(x+) 3 -2 - 2 6 6 6 6 11 EX: Graph y = cos(2x/3) 11 EX: Graph y = cos(2x/3)1

0 3 6 9 12 -1 4 4 4 411 EX: Graph y = cos(2x/3)1

0 3 6 9 12 -1 4 4 4 411 EX: Graph y = cos(2x/3)1

0 3 6 9 12 -1 4 4 4 412 EX: Graph y = 2 sin 3x 12 EX: Graph y = 2 sin 3x1

0 2 3 4 -1 6 6 6 612 EX: Graph y = 2 sin 3x1

0 2 3 4 -1 6 6 6 612 EX: Graph y = 2 sin 3x1

0 2 3 4 -1 6 6 6 613 EX: Graph y = 3 cos x 13 EX: Graph y = 3 cos x1

0 23 4 -1 13 EX: Graph y = 3 cos x1

0 23 4 -1 13 EX: Graph y = 3 cos x1

0 23 4 -1 4.2 Graphs of the Other Trigonometric FunctionsOBJ: Graph Other Trigonometric Functionsy = d + a(trig b (x + c)a (amplitude) multiply a times (0 |1 0 -1 0 1)b (period) 2 bc (starting point) d (vertical shift)Graph y = cos x03 22 214 EX: Graph y = sec x03 22 214 EX: Graph y = sec x03 22 215 EX: Graph y = 2 + sec(2x) 15 EX: Graph y = 2 + sec2(x) 2 0 2 3 4 5 6 4 4 4 4 415 EX: Graph y = 2 + sec2(x) 2 0 2 3 4 5 6 4 4 4 4 4Graph y = sin x 1

03 2 -12 216 EX: Graph y = csc x1

03 2 -12 217 EX: Graph y = csc (x + /3) 17 EX: Graph y = csc (x + /3) -2 - 0 2 6 6 6 617 EX: Graph y = csc (x + /3) -2 - 0 2 6 6 6 617 EX: Graph y = csc (x + /3) -2 - 0 2 6 6 6 618 y = tan xRef.Amp.Per. Per.

St. Pt.Vert. Sh.No140none19 y = tan (2x + /2) y = tan 2 (x + /4)Ref.Amp.Per.

Per.

St. Pt.

Vert. Sh.No128- 4none20 y = 2 + tan (x + )y=2+ tan(x + 2 )Ref.Amp.Per. Per.

St. Pt.Vert. Sh.No22-2221 y = cot xRef.Amp.Per. Per.

St. Pt.Vert. Sh.No140none22 y = 2 + cot xRef.Amp.Per. Per.

St. Pt.Vert. Sh.No14026 EX: Graph y =-3 2cos(x+5/6)6 EX: Graph y =-3 2cos(x+5/6) -5 -2 4 7 6 6 6 6 66 EX: Graph y =-3 2cos(x+5/6) -5 -2 4 7 6 6 6 6 66 EX: Graph y =-3 2cos(x+5/6) -5 -2 4 7 6 6 6 6 6